摘要：微粒群算法（PSO）是继遗传算法后的又一个基于生物演化的随机优化算法，它操作简便，收敛速度快且稳定，使得它近年来被广泛应用，并在工程实验中发挥了重要的作用。    

本文先介绍了基本微粒群算法，并在基本微粒群的基础上对其进行改进，加入了突变的部分使其能更好地收敛于全局最优解，并将改进后的微粒群算法引入到线性约束问题的最优化求解中。在研究与工程中，很多问题都带有线性约束条件，常用的方法都是先将约束问题转为无约束问题后再进行求解。传统最优化解约束问题时对问题模型有较多的条件限制，而微粒群算法对问题信息依赖度不高，因此，在问题转化为无约束后，再借助微粒群算法能更方便快捷地求解。

关键词：线性约束；优化； 微粒群算法

Abstract：Particle Swarm Optimization is another new stochastic optimization after genetic algorithm which is also based on evolution. PSO is easy to perform and convergence swiftly and steadily, such good characteristic make it be applied widely in most projects recently and it have performed an important function. 

　　　This article first describes the basic particle swarm algorithm, and on the basis of the basic particle swarm to improve it, adding some mutations enable it to better convergence to global optimal solution, and the improved PSO algorithm is introduced to linear constraints to solve the optimization problem. In research and engineering, many problems with linear constraints, commonly used methods are first constrained problem into unconstrained problem and then solve it. Best to resolve the issue of traditional constraints on the problem model has more constraints, while the particle swarm algorithm to the problem of information dependence is not high, therefore, is transformed into an unconstrained, then using particle swarm algorithm can solve more easily and quickly.

Key words: linear constraint；optimization； particle swarm algorithm

　　　微粒群算法是一种新兴的基于群体智能的进化算法。 本文介绍了算法基本思想、理论基础，以及算法的改进方法，并在一般微粒群的基础上进行改进, 并与拉格朗日法相结合, 研究了带约束微粒群算法。 通过拉格朗日对偶原理, 将拉格朗日乘子分离出来进行优化, 使得线性约束问题通过微粒群优化及少量迭代可得到最优解。该算法可以用于解决工程中带约束的问题. 最后, 通过对低通滤波器的设计, 验证了改进算法的有效性。

但是PSO算法数学基础相对薄弱,且还存在许多不完善和未涉及到的问题,尤其对离散的组合优化问题的研究还处于起步阶段。 随着微粒群算法理论研究的不断深入,应用领域必将会有更广的发展。